AN EMPIRICAL INVESTIGATION INTO THE BEHAVIOR OF OPTIONS
AROUND MERGER AND ACQUISITION ANNOUNCEMENTS
By
JAMES A. YODER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
OF F LIB
ACKNOWLEDGEMENTS
I would like to express my special thanks to the chairman of my
committee, Haim Levy, and to Drs. Roger Huang, Roy Crum and Sandy Berg.
T would also like to express my appreciation to Drs. Andy McCollough,
Craig Tapely, Robert Radcliffe, Dave Brown and Joel Houston for their
encouragement and support.
I would also like to express my gratitude to my fellow students
Young Hoon Byun, Lesa Nix, Bruce Kuhlman and Neil Sicherman for their
suggestions and technical assistance. I would also like to acknowledge
the computer programming assistance of Eric Olson.
11
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . a
ABSTRACT v
CHAPTERS
1 INTRODUCTION 1
The Behavior of Option Prices Around Merger Announcements 2
The Behavior of Implied Standara Deviations Around Merger
Announcements 4
Does the Option Market React to Merger/Acquisition
Activity Differently than the Equity Market? 5
How Does an Event Study in the Option Market Differ From
One in the Equity Marker? ......... 8
2 REVIEW OF THE LITERATURE 12
Mergers 12
Options 16
Pricing 16
Option Market Efficiency 18
Variance Bias in the Black-Scholes Model 19
3 THE BEHAVIOR OF OPTIONS AND OPTION MARKETS AROL^ND MERGER
AND ACQUISITION ANNOUNCEMENTS ... 20
Data 20
The Behavior of Options Around Merger and Acquisition
Announcements 22
The Behavior of ISDs Around Merger and Acquisition
Announcments 24
Methodology ..... 26
Interpretation Results 29
The Behavior of Call Option Prices Around Merger and
and Acquisition Announcements 34
Methodology and Results 36
Interpretation Results 40
The Behavior of Options Around Merger and Acquisition
Announcements 43
Methods and Results 47
Methods and Results 50
1X1
4 VAEIANCE BIAS AND NON- SYNCHRONOUS PRICES IN THE BLACK-
SCHOLES MODEL 60
Variance Bias in the Black-Scholes Model 61
Methodology and Results 64
Non- synchronous Prices and the Black-Scholes Model .... 71
Methodology and Results 72
Conclusion 77
5 SUMMARY AND CONCLUSIONS 79
REFERENCES 83
BIOGRAPHICAL SKETCH 87
IV
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
AN EMPIRICAL INVESTIGATION INTO THE BEHAVIOR OF OPTIONS
AROUND MERGER AND ACQUISITION ANNOUNCEMENTS
By
James A. Yoder
December 1988
Chairman: Haim Levy
Major Department: Finance, Insurance, and Real Estate
This dissertation exam.ines the behavior of options around merger
and acquisition announcements. A variation of the traditional event
study methodology was applied to the option market in order to measure
the abnormal returns accruing to the bidding firm and target firm
optionholders. The event study was extended to the equity market for
comparison purposes. The behavior of ISDs was also examined in order
to determine whether the option or equity market first reacted to the
merger/acquisition announcement and to decompose the abnormal returns
in the option market into a component due to changing stock prices and
a component due to changing stock volatilities. Some methodological
issues involving event studies were also examined.
CHAPTER 1
INTRODUCTION
Two of the most important developments in finance in recent years
have been the growth of option markets and the high level of merger and
acquisition activity. Not surprisingly, both of these areas have been
subject to intense academic scrutiny. Literally hundreds of articles
have been published on the theory and applications of options. There
are also numerous papers concerned with the rationale for mergers and
their impact on stockholders' wealth. This dissertation attempts to
relate these two subjects through an examination of oprion and option
market behavior around merger and acquisition announcements.
In order to accomplish this, four major issues will be addressed:
1. How do option prices react around merger/acquisition
announcements ?
2. How do the Implied Standard Deviations (ISDs) of options
react around merger and acquisition announcements?
3. Does the option market react to merger and acquisition
activity differently than the equity market?
A. How does an event study in the option market differ from one
in the equity market?
Each of these issues will be discussed in turn.
The Behavior of Option Prices
Around Merger Announcements
Researchers in the equity market have sought to determine whether
mergers and acquisitions produce economic gains and, if so, who reaps
the benefits. Their findings have been relatively consistent. Dodd
(1980), Asquith (1983) and Eckbo (1983), for example, have all
presented evidence on the effects of mergers on shareholders' wealth.
They conclude that most of the gains are captured by the stockholders
of the target firm. Gains to the bidding firm shareholders are small
and possibly non-existent. Their estimates of the abnormal returns
accruing to the bidding firm shareholders for the two days prior to the
announcement range from a -1.09 percent loss to a paltry 0.20 percent
gain. For the target firm shareholders, however, statistically signif-
icant gains ranging from 6.20 percent to 13.41 percent were obtained.
The merger literature is discussed more thoroughly in Chapter 2.
These results in the equity market lead to empirically testable
hypotheses for the expected behavior of options around merger and
acquisition announcem.ents . Under the assumption that the option market
!s efficient, option prices (and ISDs) can be expected to react prior
to the formal merger announcement and stabilize immediately afterward.
Merger negotiations involve many people such as investment bankers,
lawyers, administrative personnel, etc. Word of impending mergers
:!eaking to the financial market place has been amply demonstrated in
the equity market. There is no reason why the same phenomenon should
not occur in the option market.
A second hypothesis is that abnormal returns to the target firm
optionholders should exceed those of the bidding firm optionholders.
Theoretically, a call option can be duplicated by an appropriately
selected stock-bond portfolio. Because of this, the wealth effects of
merger and acquisition announcements on optionholders can be expected
to mirror that of the equityholders.
The wealth effects of merger and acquisition activity on option-
holders is of interest for a number of reasons. In a recent survey
article of the market for corporate control literature, Jensen and
Ruback (1983) identified six key questions that have been addressed.
At the top of the list is the following = "How large are the gains to
shareholders of bidding and target firms?" Options by their very
nature afford superior leverage to the underlying equity. Consequently,
optionholders, per dollar invested, have more reason to be concerned
with potential merger and acquisition activity than the equityholders.
An analysis of option prices around merger and acquisition announce-
ments may also shed light on a puzzling question.
Merger activity is widespread but the rationale for it is not
clear. As noted earlier, gains to the bidding firm shareholders are
small and possibly negative. Why then do managers undertake merger and
acquisition programs if they do not benefit the shareholders? An
examination of bidding firm option prices may help to resolve this
issue,
An option can be interpreted as a leveraged position in the
equity. This leverage aspect of options may make them more sensitive
to events than the underlying equity. Small abnormal returns in the
equity market might result in much larger abnormal returns in the
option market. Thus, it may be easier from a statistical standpoint to
determine if bidding firm stockholders benefit from merger/acquisition
activity by looking at the behavior of associated option prices.
The Behavior of Implied Standard
Deviations Around Merger Announcements
The behavior of ISDs around merger announcements is important for
two reasons. First, it is inseparable from the price behavior of
options. Stock volatility is one of the input variables for the Black-
Scholes model. By examining the ISD, it is possible to decompose
changes in option prices into a component due to price changes in the
underlying stock and a component due to changes in the underlying
volatility. Second, it provides an alternative measure of the informa-
tion content associated with merger and acquisition announcements.
Numerous studies have attempted to measure the information content of
accounting announcements by showing that the expected return of the
stock is affected. Patell and Wolfson (1979, 19Sl) have pointed out
that other moments of stock price distribution may also be affected by
the announcement and thus serve as a measure of its significance. They
proceeded to use ISDs as an ex-ante measure of the information content
associated with earnings announcements whose disclosure date is known.
This dissertation attempts to use ISDs to measure the expected impact
of merger and acquisition announcements which are totally or at best
partially anticipated events.
The hypotheses concerning the behavior of ISDs around merger and
acquisition announcements parallels that of option prices. The first
hypothesis is that ISDs should react prior to the formal announcement
and stabilize immediately afterward. The second hypothesis is that the
change in ISDs for the target firm should exceed that of the bidding
''irm options. Merger and acquisition activity has very little impact
on the bidding firm shareholders. Thus, the distribution of stock
returns should not significantly change as a result of the announce-
ment. The target firm shareholders, however, are greatly affected by
the merger and acquisition activity. Increased volatility of the
underlying stock returns can be generated by a multitude of factors.
Uncertainty, for example, can arise over the anticipated terms of the
agreement, whether a competing offer will be made or even whether the
deal will be consummated.
Does the Option Market React to Merger and Acquisition
Activity Differently than the Equity Market?
The third major area of inquiry in this dissertation is the
relationship between the option and equity markets around merger
announcements. There are two independent arguments for hypothesizing
that the merger activity will be more strongly manifested in the option
rather than the equity market prior to the announcement.
Options represent leveraged positions in the underlying equity. The
beta of an option in the Black-Scholes framework is always greater than
that of the stock. Because of this, the option market may be more
sensitive to events than the equity market. Even though the two
markets may be reacting to the same information, the signal may first
be more apparent and stronger in the option market.
It is also possible that the option market contains information
that is not incorporated in the equity market prior to mergers. As
mentioned previously, a call or put option can be duplicated by an
appropriate stock-bond portfolio. Because of this, options have been
viewed as "derivative" assets whose prices are completely determined by
the underlying equity. The possibility that the option market may
influence the equity market has received little attention. It is
conceivable that information is first processed in the option market
and then filters to the equity market. A similar issue has been
studied by Manaster and Rendleman (1982). They have advanced the
intriguing hypothesis that the option market may play a key role in
determining equilibrium stock prices. They argue that some investors
may prefer to invest in the option rather than the equity market
because of reduced transaction costs, fewer short selling restrictions,
and most importantly, superior leverage. These traders could push
option prices out of equilibrium relative to the underlying stocks.
Arbitrageurs would then intervene to restore equilibrium between the
two markets.
Manaster and Rendleman attempted to test their theory. They
"inverted" the Black-Scholes model to solve for the implied stock
price. The implied stock price was then used to predict future stock
prices. They found some evidence that the option market contains
information that is not incorporated in the equity market. Unfor-
tunately, their results are very weak and fatally flawed by their
reliance on non- synchronous data. The data used in this dissertation
will avoid this problem.
In retrospect, Manaster and Rendlemans ' lack of results is not
surprising. Both the option and equity markets react to public
information. Generally, one would expect both markets to adjust
simultaneously to new public information. On any given day for any
particular corporation there may not be and probably is not information
that is not fully reflected in both markets.
However, this may not be true prior to major announcements by
corporations such as mergers. In the case of mergers, the option
market could be expected to be particularly influential in determining
stock prices around merger announcements. Keown and Pinkerton (1981)
have argued that information concerning impending mergers is suscep-
tible to insider exploitation due to the large number of people
typically involved in the negotiating process. They have attributed
increased trading volume before the merger announcement to insider
activity. An insider attempting to profit from knowledge of an impend-
ing merger would have an incentive to use options because of their
leverage aspects. To quote Fischer Black (1975, p. 61), "Since an
investor can usually get more action from a given investment in options
than he can by investing in the common stock, he may choose to deal
with options when he feels he has an especially important piece of
information." Option prices can be expected to contain more informa-
tion than the equity market if nonpublic information is being
exploited. If information regarding future mergers first reaches the
financial markets through insider trading in options, merger activity
may very well be reflected in the option market before the stock
market.
A separate issue raised by the above argument is that the option
market m.ay make the equity market more efficient. If the option market
serves to bring nonpublic information into the financial markets and
options influence the prices of the underlying stock, then stocks with
listed options should respond sooner to impending m.ergers than similar
stocks without options.
How Does an Event Study in the Option Market
Differ From One in the Equitv Market?
An event study attempts to measure the impact of some event on
securityholders by comparing actual returns around the announcement to
those predicted by some model. These predicted returns should be the
returns that would have occurred if the event (merger and acquisitions
in this case) had not taken place. The difference between the actual
and predicted returns is the basic measure of the impact of the event.
These residuals are then aggregated to measure the total impact of the
event and provide statistics for tests of significance.
In the equity market, predicted returns are usually generated by
one of two models:
a. Market model
b. Mean returns
The market model assumes there is a linear relationship between
individual security returns and market returns:
Kit = a^ + 3iRmt
Where R^^ = return on company i for day t
-R-nit = return on the market for day t
The coefficients a and are obtained by regressing the company
returns against the market returns over some base period prior to the
merger activity. The base period should be selected so that the
company returns are not affected by the event activity.
.'\nother approach to generating predicted returns is to simply use
the mean returns on the individual security computed over some base
period
_ n
R^ = 1/N 2 R,;t
where R^ = mean return on company i
Rj_f = return of company i for day t
N = number of observations in the base period
Again the base period should be selected so that the event activity has
no effect.
Two implicit assumptions underlie the traditional event study
methodology in the equity market. The first is that the return
generating process is linear. As long as predicted returns equal actual
returns on average, the residuals should average out to zero over a
large enough cross-sectional sample in the absence of some conimon
disturbing event. The same reasoning justifies parameter estimation
for the two models. The true beta is unknown and must be estimated.
The estimated beta may lie above or below the true value. As long as
an unbiased of beta is used, however, deviations from the true beta
return will average out to zero. Since these models are linear,
deviations from the true expected return will also offset and residuals
should average out to zero in the absence of a common disturbance. The
second assumption is that the return generating process is stationary.
Specifically, beta is assumed to remain constant over time.
Call prices in a Black-Scholes framework are a function of five
input variables. Two of these, the stock price and its volatility, are
company specific and would be affected by an event such as a merger
acquisition announcement. One implication of this is that there may be
10
a subtle but important difference between the interpretation of the
results of an event study in the option and equity markets.
i-!e.tger activity may not benefit a stockholder even if abnormal
returns are observed. These abnormal returns may be accompanied by
increased risk. This increased risk may not be desired by an investor
with a small portfolio even if it is compensated for by larger expected
(not realized) returns.
If an investor holds a call option, the situation may be entirely
different. An increase in the volatility of the underlying srock would
definitely be preferred by all investors. Increased volatility would
result in an actual (not expected) increase in the call price. The
reason for this is that the return generating process underlying call
prices is based on the formation of risk-less hedged portfolios.
The Black-Scholes formula is by far the most widely used option
pricing model. Using it to generate predicted returns for an event
study, however, presents some technical problems. The Black-Scholes
model is highly non-linear. Consequently, using sample estimates for
the input variables may result in a systematic bias. Errors in estimat-
ing the variables may offset in a large sample. Equal deviations from
the true parameter estimate, however, will not result in equal devia-
tions from the true call price. The most crucial variable is the stock
volatility since the Black-Scholes model is most sensitive to it.
Because of this problem, the results of an event study utilizing
the Black-Scholes model must be interpreted with care. A simulation
analysis, however, provides some m.easure of the magnitude of this
effect. The Black-Scholes formula was used to generate a theoretical
option price assuming true values for the input parameters. Sample
11
estimates of the volatility were generated for input into the Black-
Scholes model. These sample call prices based on sample estimates for
the volatility were compared to the theoretical call value. In
general, the difference was small (see chapter A).
CHAPTER 2
REVIEW OF THE LITERATURE
The literature on option theory and mergers, as mentioned previ-
ously, is immense. It is impossible to discuss in detail all the
relevant studies in either of these fields. At best, the most impor-
tant results can only be highlighted. This section will give a brief
review of the work that directly affects this dissertation. The
literature dealing with the impact of mergers on shareholders wealth,
option pricing, option market efficiency, and variance bias in the
Black-Scholes model will be addressed in turn.
Mergers
Two fundamental questions have been raised regarding merger
activity. The first is why do mergers occur? In 1985 alone, merger
activity involved over $120 billion in assets. Yet the economic
justification for all this activity is not obvious. Levy and Sarnat
(1970) have shown that given perfect capital markets, pure conglomerate
mergers should not create value.
Agency theory provides one rationale for the continuous merger
activity that has been observed over the past few decades. Levy and
Sarnat (1970), Lewellen (1971) and Gali and Masulis (1976) have argued
that combining firms with less than perfectly correlated cash flows
lowers the chances for bankruptcy. Thus, managers have an incentive to
engage in merger activity so as to reduce their employment risk. Reid
(1968) has argued that managers strive to maxim.ize the size of the firm
12
13
rather than shareholder wealth. Jensen and Heckling (1976) have pointed
that since managers are agents for the stockholders' their interests
are not necessarily the same.
Others have sought to justify merger activity on the grounds that
it produces real economic gains. Mergers may result in more efficient
economic units. Weston and Chung (1983) have summarized possible
sources of these efficiencies.
1. Differential Efficiency
2. Inefficient Managem.ent (target firm)
3. Operating Synergy
A. Financial Synergy
5. Strategic Realignment
6. Undervaluation (target firm)
-..- of now, however, the exact rationale for mergers is still an
unresolved issue.
The second major issue that the merger literature has addressed is
what are the effects of mergers on shareholders' wealth? Numerous
studies concerned with this issue have appeared since Mandleker's
(1974) seminal paper. Most of these have used the well known event
study methodology.
Event studies in the equity market involving mergers have become
relatively standardized. A base period prior to the event is selected,
and data from this period are used to estimate predicted returns. The
impact of the event is measured by calculating the difference between
the actual and predicted returns during some period surrounding the
event. The residuals are then aggregated and statistically analyzed,
14
usually using some form of a t-test, to determine if the excess returns
are significantly different from zero.
Predicted returns in the equity market have usually been generated
by one of two models. The first method is to use the market model.
Rjt = aj + Bj"R^t
The estimates of aj and Bj are obtained by regressing the company
returns against the market returns during some base period prior to and
presumably untainted by the merger activity. The second method is to
simply use the mean return computed over some base period.
Brown and Warner (1985) have shown that the event study metho-
dology is very robust to the method used to calculate excess returns.
Using simulation analysis, they showed that there is very little
difference in the returns (or residuals) generated by the two methods.
Because of this, mean adjusted returns will be used in this study.
Jensen and Ruback (1983) have summarized the results of the more
important merger studies concentrating on announcement effects. These
results are shown in the table below. The top panel shows the results
for the two days prior to the announcement. The bottom panel shows the
results for the one month prior to the announcement. In each case, the
total return during the event period, the number of observations and
the t-statistic is given for both the bidding and target firms.
The results are very consistent. The gains to the acquiring
firms are positive but small. The target firm stockholders reap much
larger returns. This is true for both the short-term (2 day) and long-
term (one month) event period. In addition, these results hold for
both successful (consummated) and ultimately unsuccessful mergers.
15
Table 2.1
A'Dnormal Returns Associated with Mergers;
Sample Size and t-statistic
Study Sample Bidding firm Target firm
period
A. Two-day announcement effects
T)odd /0-77 -1.09" +13.41
(1980) (60"", -2. 98""'") (71^23.80)
Asquith 62-76 +0.20 +6.20
(1983) (196,0.78) (211,23.07)
Eckbo 63-78 +0.07 +6.24
(1983) (102,-0.12) (57,9.97)
Weighted excess return -0.05 +7.72
B. One-month announcement effects
Eodd 70-77 +0.80 +21.78
(60,0.67) (71,11.93)
Asquith 62-76 +0.20 +13.30
(1983) (196,0.25) (211,15.65)
Eckbo 63-78 +1.58 +14.08
(1983) (102,1.48) (57,6.97)
Asquith et. 63-79 +3.48 +20.5
al. (1983) (170,5.30) (35,9.54)
Malatesta 69-74 +0.90 +16.8
(1983) (256,1.53) (83,17.57)
weighted excess return +1.37 +15.90
-excess return
""'number of observations
" " "t - Stat is t i c
16
The paper by Asquith, Bruner and Mullins deserves additional
comment. Schipper and Thompson (1983) have shown that acquisition
programs generate excess returns. If this is true, one might expect
the impact of successive mergers to diminish. Asquith et. al. compare
the abnormal returns associated with the first, second, third, and
fourth mergers. They find no evidence that abnormal returns are
capitalized in the earlier mergers. They also found that the abnormal
returns to the acquiring firm is dependent on the size of the acquired
firm.
Options
Pricing
The seminal work on option pricing is, of course, the Black-
Scholes option pricing model. Black and Scholes (1973) noted that a
call and the underlying stock could be combined to form a risk-free
hedged portfolio if continuous rebalancing was possible. This fact,
combined with some appropriate assumptions
1. frictionless capital markets
2. risk-free interest rate is constant
3. stock pays no dividends
A. stock prices follow an Ito process with constant drift
5. no restrictions on short sales
allowed them to derive a differential equation relating call and stock
prices. Using stochastic calculus, they solved for the call price
yielding the familiar Black-Scholes formula as a result
C = SN(d^) - Xe-^T^J(^i^)
where
17
d;, = [ln(S X) + (r + 0.5o-)T]/ajT
d2 = dT_ - aVT
The most limiting of the Black-Scholes restrictions is that the
underlying stock pays no dividends. Modifying the model for dividends
has two components. First, the stock price must be adjusted for the
expected drop on the ex-dividend date. Second, the model must reflect
that an American call has value due to its early exercise right. If a
dividend is large enough, it may pay to exercise the option immediately
before the stock goes ex-dividend. These problems can be dealt with
simply by subtracting the present value of future dividends from the
stock price as Black (1975) has suggested or assuming that dividends
are paid continuously as Merton (1973) has done. Roll (1977), Geske
(1979b), and Whaley (1981) have advanced more complex formulation that
take both considerations into account. Whaley (1979) has empirically
tested the different approaches to dividend adjustment and found the
differences were slight.
A number of variants and extensions of the Black-Scholes model
have appeared. Merton (1973) has relaxed the assumption of stationary
interest rates. Thorpe (1973) has examined the effect of short sales
restrictions. Geske (1979a) has developed a compound option formula.
The effects of different distributional assumptions regarding
stock prices have also been investigated. Cox and Ross (1976) have
developed a pure jump model that allows for discrete stock price
movements. They have also developed a constant elasticity of variance
model that allows for the variance to change with the stock price.
Merton (1976) has developed a mixed diffusion- jump model that super-
imposes a jump process on a continuous return generating process.
18
Despite these advances, the Black-Scholes model with the stock
prices adjusted for dividends is still the most widely used by far.
Many of the models discussed above are difficult if not impossible to
apply. Even if they can be applied, no model has yielded consistent,
significantly better results for options near the money. While there
are limitations to the Black-Scholes model, there is no strong reason
to use any of the more esoteric alternatives in this study.
Option Market Efficiency
A number of studies have been made of the efficiency of the
Chicago Board of Options exchange. These studies are joint tests of
market efficiency and the Black-Scholes model. Galai (1977) conducted
one of the earliest and most comprehensive studies using the Black-
Scholes model to identify mispriced options. He found that statisti-
cally (but not economic) significant excess returns could be earned.
Chiras and Manaster (1978) adopted a different approach in
analyzing option market efficiency. They weighted the ISD of each
option on the stock by the option price elasticity to arrive at an
overall measure of the stock's future volatility (WISD) . They then
compared the WISD as an estimate of future stock volatility to esti-
mates based on past stock returns. Having demonstrated the superiority
of WISDs, they then proceed to compute imiplied option prices. Under-
priced and overpriced options were then identified by comparing implied
and actual prices. Risk-free hedges were then formed which earned
substantial abnormal returns. These results are in agreement with a
similar study by Trippi (1977) which used a simpler weighting scheme to
arrive at WISDs. Kalay and Subrahmanyara (1984) have also provided some
19
evidence of option market inefficiency on the ex-dividend day of the
equity.
Phillips and Smith (1980) have found fault with studies reporting
inefficiencies in the option market. They argued that a close examina-
tion of trading costs (most notably the bid-ask spread) would account
for the abnormal returns reported in earlier studies. Bhattacharya' s
(1980) study of CBOE (Chicago Board of Options Exchange) took these
costs into consideration. In general, his results were consistent with
market efficiency.
Variance Bias in the Black-Scholes Model
In order to apply the Black-Scholes model, five input variables
must be obtained: the stock price, exercise price, time to maturity,
risk-free rate of interest and the volatility of the underlying stock.
Of these, four are directly observable. Only the variance of the
underlying stock returns needs to be estimated.
Classical methods of estimating the variance will bias the model.
Although unbiased estimators of the variance exist, the Black-Scholes
model is highly non-linear. Equal deviations from the true variance
will not result in equal deviations from the true call price as
Ingersoll (1977) and Merton (1975) have observed. Boyle and
Ananthanrayanan (1977) have used numerical integration to examine the
magnitude of the expected error in a single case. Butler and Schachter
(1986) trace the behavior of this bias to the second derivative of the
cumulative normal density function.
CHAPTER 3
THE BEHAVIOR OF OPTIONS AND OPTION MARKETS
AROUND MERGER AND ACQUISITION ANNOUNCEMENTS
This chapter discusses the behavior of options and option markets
around merger and acquisition announcements. The impact of merger and
acquisition announcements will be studied by examining the behavior of
option prices and ISDs around the announcement date. These results
will be compared to those obtained from the underlying equity using the
traditional event study methodology.
The organization of this chapter is as follows. First, the data
is described and a potential problem discussed. Next, the behavior of
options around merger and acquisition announcements is analyzed. The
return of an option is affected by two company specific variables: the
stock price and stock volatility. Both of these variables are likely
to be affected by merger and acquisition activity and thus influence
call returns. An attempt is made to decompose the total impact of
merger and acquisition activity into two components. First, the effect
of changing ISDs is investigated and then the total impact due to both
changing stock prices and stock volatility is analyzed. Finally, the
traditional event study methodology is applied to the underlying stock
in order to compare the behavior of the two markets.
Data
The merger and acquisitions selected for this srudy will be
obtained from Mergers and Acquisitions . The mergers and acquisitions
20
21
selected will be confined to those involving at least $100 million in
assets with either the acquiring or acquired firm having options listed
on the CBOE between 1982 and 1985. The reason for this is to ensure
the merger and acquisition is an event. Corporations listed on the
CBOE tend to be well established firms with large equity bases. The
value of all the outstanding stock in firms such as General Motors,
General Electric and International Business Machines, for example, is
measured in the billions of dollars. The announcement date will come
from the Wall Street Journal Index.
The Wall Street Journal will also be used to get the bid-asked
spread on U.S. Treasury bills in order to calculate the risk-free
rate. The risk-free rate for input into Black-Scholes formula will be
the yield on the T-bill maturing closest to the expiration date of the
option. The yield will be calculated according to the formula below
from Cox and Rubinstein (1985, p. 255)
r = (F/lG,GOO)'l/t
where r = one plus the risk-free rate
P = price of a $10,000 T-bill
= 10,000 [l-0.01(bid + asked)/2 (n/360)]
n = number of days to maturity
t = time to maturity expressed in years
The critical data for this thesis is the stock and option prices.
Closing prices from the Wall Street Journal or similar sources can not
be used because of the possibility of nonsynchronous trading between
the two markets. Trading in the option market is significantly less
active than in the equity market. It is quite likely that the last
22
option trade occurred prior to the last stock trade on any given day.
The current stock price for use in the Black-Scholes formula is the
price existing at the time the option is being valued. Using the
Black-Scholes model to generate predicted returns as in this study
requires the stock price at the time of the last option trade. This
problem, as Bookstaber (1981) has pointed out, casts doubt on much of
the empirical work on options that has been done to date.
In order to avoid any problems with nonsynchronous trading,
time-stamped data will be used. The Berkeley Options Tape will be the
primary source of stock and option price information. Data from
Francis Emory Fitch, Inc., although not machine readable, is also
suitable.
The Behavior of Options Around Merger
and Acquisition Announcements
The behavior of stocks around merger announcements has been
extensively studied (see chapter 2). The results have been very
consistent. Most of the gain due to merger activity is captured by the
stockholders of the target firm. Gains to the bidding firm share-
holders are small and possibly non-existent.
These results suggest empirically testable hypotheses for the
expected behavior of options around merger and acquisition announce-
ments. Market anticipation of formal merger announcements has been
observed in the equity markets. Under the assumption that the option
market is efficient, option prices and ISDs should react prior to the
formal merger and acquisition announcement and stabilize immediately
afterward. A second hypothesis is that the abnormal returns to the
target firm optionholders should exceed those of the bidding firm
23
optionholders. Theoretically, a call option may be duplicated by an
appropriately selected stock-bond portfolio. Because of this, the
wealth effects of merger and acquisition announcements can be expected
to mirror that of the equityholders. This assumes, however, that other
factors such as the stock volatility are not affected by the merger and
acquisition activity.
There are two main reasons why an analysis of the impact of merger
and acquisition activity of optionholders wealth is of interest.
First, the methodology used, the traditional event study, has never
been applied to the option market. An event study in the option
market presents new issues and sheds light, as will be discussed later,
on event studies in the equity market. The second major reason why the
wealth effects of merger and acquisition activity is important is that
options afford superior leverage to the underlying equity by their very
nature. Optionholders, per dollar invested, have more reason to be
concerned with the effects of merger and acquisition activity than the
equityholders .
These ideas will be more fully developed later on. At this point,
i
1 the effect of merger and acquisition on option ISDs will be discussed.
The behavior of ISDs around merger and acquisition announcements is not
I only a major determinant of call option returns and thus optionholders
wealth but also has important implications for event studies in the
equity market .
Event studies in the equity market implicitly assume that risk
1 remains constant. Predicted returns are based on historical data from
some base period. Increasing ISDs (i.e., stock volatility) would
suggest that risk is increasing. More importantly, since ISDs can be
24
computed at a point in time and are correlated with stock betas, they
can be used to adjust for increasing risk in the event period. This is
discussed later on in this chapter.
The Behavior of ISPs Around Merger
and Acquisition Announcements
The behavior of option ISDs around merger and acquisition
announcements is important for a variety of reasons. First, it is
inseparable from the price behavior of options. Stock volatility is
one of the input variables for the Black-Scholes model. By examining
the behavior of the ISDs it is possible to decompose changes in option
prices into a component due to price changes in the underlying stock
and a component due to changes in the underlying volatility.
A second reason for examining the behavior of ISDs is that it
provides an alternative measure of the information content associated
with merger and acquisition announcements. The vast majority of event
studies have attempted to measure the information content of some event
by showing the expected return of the stock is affected. Patell and
Wolf son (1979,1981) have pointed out that other moments of the stock
price distribution may also be affected and thus serve as a measure of
its significance. They used ISDs as an ex-ante measure of the informa-
tion content associated with earnings announcements whose date is
known .
This study will determine if the second moment (stock volatility)
of the stock return distribution is affected by merger and acquisition
activity. The behavior of ISDs will be tracked around the announcement
date for both the bidding and target firms in order to determine if
there is a difference in the impact of the activity between the two
25
categories. It should be noted that merger and acquisition announce-
ments are unexpected or at best partially anticipated. This fact
distinguishes this study from the ones by Patell an Wolfson which dealt
with earnings announcements on known dates.
Another reason for examining the behavior of ISDs is that it may
shed light on potential wealth shifts engendered by merger and acquisi-
tion activity. Option theory suggests that common stock can be
interpreted as an option. Agency theory suggests that there is an
incentive for stockholders (see Jensen and Heckling 1976) to shift
wealth from the bondholders by undertaking risky investm.ent projects.
By undertaking investment projects which increase the variability of
the firm's cash flows, the stockholders' can, in effect, gamble with
the bondholder's money. This enriches the stockholders at the direct
expense of the bondholders. Merger and acquisition activity can be
regarded just like any other investment activity. Consequently, one
might expect bidding firms to make acquisitions which tend to increase
the variability of the firm's cash flows.
Others, however, have argued that the opposite occurs. Levy and
Sarnat (1970), Lewellen (1971) and Galai and Masulis (1976) have argued
that combining the cash flows of two independent com.panies may reduce
the probability of default and increase the market value of debt at
the stockholders' expense. Even if this occurs, it is possible that
managers act to neutralize (issue more debt) any such wealth shift.
In any case, it is the variability of the firm's cash flow that is
in question. A direct relationship, however, has been hypothesized in
previous work (see Eger 1983). Consequently, the behavior of ISDs
26
around merger/acquisition announcements may be of value in analyzing
whether these wealth shifts do, in fact, take place.
■Methodology
The methodology used for analyzing ISD behavior is as follows.
First the sample was stratified into two groups. The first group was
composed of 52 bidding firms involved in a merger or acquisition. The
second group was composed of 21 target firms involved in a merger or
divestment.
Base ISDs were obtained by "inverting" the Black-Scholes model
using data forty days prior to the announcement date. It is assumed
that the markets have not yet begun to reflect the merger and acquisi-
tion activity at this point. If the 40th day prior to the announcement
is a holiday or weekend, the first trading day afterwards is used.
Dividends are assumed to be paid continuously and are adjusted for as
suggested by Merton^ (1973). The stock and option prices are the first
prices from the Berkeley option tapes after the stock price has changed
once. The opening trade is eliminated in order to ensure the market
has stabilized. ISDs are calculated in a similar manner for each
company for each day in the event period. The event period ranges from
five days prior to the announcement date to two days afterward. It
should be noted that the announcement day is taken to be the date it
first appeared in the Wall Street Journal . In many instances, the news
was released during trading hours of the previous day.
Dividends are adjusted for by using Merton's (1973) formula
C = Se"ytN(di) - Xe'^^Nid-y)
where dl = [ln(S/X^ + (r - y - 0T5a2)t]/aVt
d2 = dl - oVt
y = continuous dividend vield.
27
The impact of the merger and acquisition activity on option ISDs
was measured by taking the difference between the ISD for each company
for each day during the event period and the base ISD for each company
SISDjt " ISDjt - ISDbj
where 6ISD^^ = Change in ISD for company j on day t.
(t = -5 to +2)
ISD-^ = ISD for company j on say t
ISDb^ = Base ISD for company j
A t-test was run on the change in ISDs for each day in order to
determine statistical significance:
t = 6ISDj/(s2/N)"l/2
The results are given in Table 3.1. For each day in the event
period the mean change in the ISD is given, the t-statistic and the
probability (if significant at the lOZ level) of exceeding the absolute
value of the t-statistic given there was no change in the distribution
of ISDs between the base and event periods.
The effect of changing ISDs on call prices was also investigated.
For each company, for each day during the event period, the closing
price of the option closest to the money with at least 30 days to
maturity was obtained. The Black-Scholes Model was used to compute a
call price on the same option using the base ISD but actual (market)
stock prices. The percentage difference between the actual (market)
call price and the theoretical base price obtained using base period
ISDs in the Black-Scholes model was calculated for each company for
each day in the event period
28
Table 3.1
Average Change in ISDs Between the Base and Event Period
Bidding Firms
ay
Average
t-
Prob >
ISD Change
Statistic
It!
-5
-0.010678
-1.41
-4
-0.009359
-1.46
— s
-0.002800
-0.39
-2
0.000765
0.12
-1
-0.001085
-0.17
-0.005835
-0.73
+1
-0.005354
-0.58
+2
0.000932
0.10
Target Firms
Day
Average
t-
Prob >
ISD Change
Statistic
/t/
-5
0.025402
1.35
-4
0.021308
1.34
-3
0.033090
1.92
.0690
-2
0.034044
2.01
.0580
-1
0.048052
3.03
.0071
0.056132
2.39
.0266
+1
0.024384
1.53
+2
0.004447
0.24
29
where %&C^^^ = % Difference between the actual and the base
call price for company j on day t
C-!^ = Actual market call price for company j on day t
Cb^|- = Base price obtained from using base ISD in the
Black-Scholes Model for company j on day t.
Since the observed (market) stock price is used to obtain the base call
price (Cb^-j-), the difference between the actual and base call prices
must be entirely due to the changing stock volatility.
A t-test was run on the percentage deviation from the actual call
prices in order to determine statistical significance
t = %6Cjt / (s2 / N)"l/2
The results are given in Table 3. For each day in the event period,
the mean percentage deviation is given, the t-statistic and the
probability (if significant at the 10% level) of exceeding the absolute
value of the t-statistic assuming there was no change between the base
and actual market call prices.
I-'-erpretation of Results
As one might expect, the above two tables are very consistent.
They may be regarded as opposite sides of the same coin. The change in
ISDs for the bidding firm is small and statistically insignificant (at
the 10% level) in all instances. Similarly, the percentage deviation
of market prices from base prices is also small and statistically
insignificant. The change in ISDs for the target firm are much larger
than those of the bidding firm for corresponding days in the event
period. Furthermore, the change is always positive and statistically
30
significant for days -3 through the announcement date (Day 0). The
same observations hold for the percentage deviation in prices.
These results are consistent with the hypotheses of option market
efficiency. For the bidding firm there is no evidence that the merger
and acquisition activity has any effect on the volatility of the
underlying stock. The change in ISDs are very small and do not result
in large, statistically significant changes in the call prices. There
does not appear to be any changes in the ISDs or call prices before and
after the merger and acquisition announcement. The target firms are
definitely affected by the merger and acquisition acrivity. The
average change in ISDs is over 5 percentage points in absolute terms on
Day and is responsible for call price increases of ovex 12%. The
market, however, starts to anticipate the merger and acquisition
announcement as early as three days ahead of time. The change in ISD
from the base level jumps from roughly 0.021 on day -4 to 0.033 on day
-3.0 to -0.048 on day -1 to 0.056 on day 0. The percentage change in
call prices follow a similar pattern. Immediately after the announce-
ment is made public, however, ISDs and call prices quickly stabilize at
close to their base levels. The deviation of the market from the base
call price is only 0.009 for the target firms the day after the
announcement .
The results in Tables 3.1 and 3.2 also support the hypothesis that
ISDs can be used to measure the information content of merger and
acquisitions announcements. Studies in the equity market have shown
that mergers do not greatly affect the expected return of the bidding
firm stockholders. It would appear that the volatility of returns is
31
Table 3.2
Percentage Deviation Between Market and Base Call Prices
Day
Bidding Firms
% Deviation t-Statistic
Prob > /t/
-5
-4
-3
-2
-1
+1
+2
-0,009791
-0.016577
-0.058152
0.002362
-0.030124
■0.021935
-0.021193
■0.013851
0.48
-0.78
-1.11
0.12
-1.49
-1.05
-0.89
-0.55
Target Firms
Day
% Deviation
t- Statistic
Prob > Itl
-5
0.035339
0.76
-4
0.037151
0.85
-3
0.067024
1.66
0.1133
-2
0.085629
1.74
0.0966
-1
0.126755
3.14
0.0056
0.058533
1.65
0.1146
+1
0.008906
0.10
+2
-0.047437
-0.83
32
also unaffected. Changes in the bidding firm ISDs are very small and
statistically insignificant.
Studies in the equity market have also shown that significant
abnormal returns accrue to the target firm shareholders. The results
here indicate these abnormal returns are accompanied by increased
return volatility. Tt should be noted that the numbers in Table 3.1
are absolute changes from the base ISD. The percentage deviations from
the base ISD would be much larger.
Why does merger and acquisition activity have such a major impact
of the second moment (variance) of the return distribution of the
target firms? As mentioned earlier, results in the equity market have
shown that most of the gain from merger activity is captured by the
target firm shareholders. The rationale for this is that the takeover
market is competitive. If a company has some unique aspect that other
companies can exploit, it will find or have the potential to find a
number of bidders. Competition among the bidding firms will drive the
net present value of the investment to zero (see Mandelker (197A) and
Jensen and Ruback (1983)). Consequently, the gains from merger and
acquisition activity will be reaped by the target firm shareholders.
Because of this, merger and acquisition activity could be expected
to affect the volatility of the target firm's equity much more then
that of the bidding firm's. Merger and acquisition activity is a more
or less neutral event for the bidding firm shareholders. Target firm
shareholders, however, are likely to be greatly affected. The impor-
tance of merger and acquisition activity to the target firm share-
holders combined with uncertainty over the terms of the agreement,
whether alternative bidders will appear, whether the agreement will be
33
consummated, etc., should result in higher ISDs for the target firm
options.
Table 3.2 confirms this hypothesis. The average change for the
bidding firms is negligible. The average change in ISD for the
target firms is larger than that of the bidding firms for the corre-
sponding day in all cases. In some instances, the change in the
target firms' ISDs exceeds those of the bidding firms by more than an
order of magnitude.
This result warrants further comment. Event studies in the
equity market have demonstrated time and time again that target firm
shareholders reap abnormal returns as a result of merger activity.
These abnormal returns, however, are accompanied by increased vola-
tility as Table 3.2 shows. Thus, these "abnormal returns" may not
truly be abnormal but merely reflect the increased uncertainty and
riskiness engendered by the merger and acquisition activity. Instan-
taneous or short-term adjustments for risk are difficult in the equity
market since beta requires historical time series to estimate. The ISD
of an option, however, can be determined at a point in time. Thus, an
event study in the option market may afford a better measure of excess
return. This point is explored more deeply in the next section.
A final comment on the behavior of ISDs. It is interesting that
the ISDs for both the bidding and target firms' revert back to their
base level after the merger and acquisition announcement (see Table
3.2). It would appear that merger and acquisition activity does not
result in permanent changes in the volatility of the underlying equity
for either bidding or target firms. The evidence here does not support
the hypotheses that wealth shifts between bondholders and stockholders
34
arises from merger and acquisition activity. This is counterintuitive.
One would expect the post-announcement ISD to be function of the
volatility of the underlying equity of both companies and their
correlation. These results may be due to the sample selection process.
Companies listed on the CBOE tend to be large com.panies. Thus, a
merger and acquisition of $100 million may still be insignificant. In
addition, large mergers or acquisitions are likely to take place
between solid, established companies of relatively equal size. Thus,
any conclusions concerning wealth shifts and merger and acquisition
activity based on the data here must be interpreted with great care.
The Behavior of Call Option Prices Around Merger and Acquisition
Announcements
Many of the reasons for examining the behavior of option prices
around merger and acquisition announcements have already been discussed
earlier. First, it provides a test of option market efficiency.
Merger (and acquisition) negotiations involve many people such as
investment bankers, lawyers, administrative personnel, etc. Word of
impending mergers leaking to the financial market place has been amply
demonstrated in the equity market (see Keown and Pinkerton, 1981).
There is no reason why the same phenomenon should not occur in the
option market.
The price behavior of options around merger and acquisition
announcements is important to anyone who intends to invest or speculate
in options. Merger and acquisition activity is a major economic factor
in our economy and is likely to remain so for some time. Anyone
involved in options may be confronted with an unanticipated merger or
acquisition announcement. Options by their very nature afford superior
leverage to the underlying equity. Optionholders, per dollar invested.
35
are more affected by merger and acquisition activity than the equity-
holders. In order to invest intelligently, potential optionholders (or
sellers) need to have some idea of how merger and acquisition activity
could potentially affect their wealth position.
Another reason for analyzing the behavior of option prices around
merger and acquisition announcements is that it may help to determine
if this type of activity is an "event" from the standpoint of the
bidding firm. As noted earlier, gains to the bidding share holders are
small, possibly negative and statistically insignificant (see Chapter
2). It also appears that merger and acquisition activity has no effect
on the volatility (second moment of the return distribution) of the
bidding firm's equity. It is possible, however, that the option
prices of the bidding firms might still measurably react to the merger
and acquisition activity.
An option can be interpreted as a leveraged position in the
equity. The leverage aspect of options may make them more sensitive to
events than the underlying equity. A shock or event that provides an
insignificant abnormal return in the equity market might be magnified
into an identifiable, significant abnormal return in the option market.
Thus, it might be easier from a statistical standpoint to determine if
merger and acquisition activity is an event to the bidding firm
security holders.
A final reason for examining the price behavior of call options is
that this is the first study to apply the traditional event study
methodology to the option market. The event analyzed here is merger
and acquisition announcements. The methodology employed, however, has
36
general applicability. It can be applied to any event such as dividend
or earning announcements.
Methodology and results
An event study attempts to measure the impact of some event on
securityholders by comparing the actual, observed market returns to
those predicted by some model. Ideally, these predicted returns should
be the returns that would have occurred if the event (merger and
acquisition activity) had not taken place. This study uses the Black-
Scholes model to generate predicted returns.
In a Black-Scholes framework, call option prices change when
there is a change in the risk-free rate, the time to expiration, the
exercise price, the stock price or stock volatility. In equilibrium,
the actual, observed market price equals the theoretical, Black-Scholes
price. Here the announcement effect is measured by the impact of the
changing stock price and volatility on the option price. The observed
call option price is compared to a predicted price generated by the
Black-Scholes model that keeps the stock price and volatility constant.
An event study in the option market is fundamentally different
from the one in the equity market. Event studies in the equity market
assume the return generating process is linear and that the true beta
remains constant over time. As long as predicted returns equal actual
returns on average, residuals should average out to zero over a large
enough cross-sectional sample in the absence of some common disturbing
event. This also justifies using an estimate of beta. The true beta
is unobservable and must be estim.ated. If an unbiased estimate of beta
is used, deviations from the true expected return will also offset and
37
residuals should average out to zero in the absence of a common
disturbance.
The return generating process in the option market, however,
makes an event study inherently different from one in the equity
market. A cursory examination of the Black-Scholes model shows it is
highly non- linear. Even if unbiased estimators are used to obtain
inputs for the model, equal deviations from the true parameters will
not result in equal deviations from the true call price. Thus,
residuals will be biased simply due to the estimation of the input
variables. This issue has important implications for users of the
Black-Scholes model and is analyzed at length in Chapter 4.
Another difference between an event study in the two markets is
that the uncertainty of an option is an explicit function of time. The
uncertainty of an option with a short time to maturity is greater than
the same option with a longer life. The Black-Scholes model incor-
porates this time dependency and will be used for this study.
The methodology used to examine the behavior of call prices
around merger and acquisition announcements is as follows. First, as
before, the sample was divided into bidding and target firms. A
number of options with different exercise prices and maturities exist
for a company on a listed exchange. One option was selected to avoid
statistical dependence in the returns. The exercise price selected was
the closest price to the stock price forty days prior to the announce-
ment. It is assumed that the impending announcement will not be
reflected in option prices at this point. The maturity selected will
be the first expiration date at least thirty days after the event
period.
38
The reason for these particular choices of exercise price and
maturity is to mitigate problems with using the Black-Schoies formula.
The Black-Scholes formula has been found to be less accurate for deep
in-the-money or out-of-the-money options. Thus, an option near-the-
money is used. The reason for insisting the option have at least 30
days to expiration is that the Black-Scholes model has been shown (see
Manaster and Rendleman 1982) to be sensitive to its underlying assump-
tions for options close to expiration.
Once an option is selected according to this criteria, returns
will be computed for each day in the event period. These returns will
be matched with predicted returns computed from prices generated by the
Black-Scholes formula.
The predicted returns should be untainted by the merger and
acquisition activity. Of the five input variables for the Black-
Scholes model, only the stock volatility and stock price are likely to
be affected. The obvious approach to estimating the stock volatility is
to use historical stock returns from some base period. Another method
is to "solve" the Black-Scholes formula for the implied standard
deviation. ISDs reflect market expectations and should provide better
estimates of future stock volatility than historical data. This has
been confirmed by Latane and Rendelman (1976), Trippi (1977) and Chiras
and Manaster (1978). Although a number of complex weighting schemes
have been suggested, Beckers (1981) has demonstrated that using the ISD
from the option nearest the money may work just as well. For this
reason, the base ISDs computed in the previous section will be used to
proxy the base volatility. The efficient market hypothesis suggests
that the best estimate of tomorrow's stock price is today's stock
39
price. For this reason, the closing stock price 40 days prior to the
announcement date (which is assumed to be unaffected by merger and
acquisition activity) is used as the input stock price.
Residuals will be computed for each company for each day in the
event period
"j.t = Rj,t - R'^j,t
where U^,^ = residual for company j on day t
J
R
j,|- - actual (observed) option return for company
j on day t
R*jjt ~ predicted option return for company j on day t
Next daily average residuals will be com.puted to measure the impact of
the merger and acquisition announcement for each day in the event
period
_ n
U^- = 1/N .E,U. ^
^ j=i J>t
where U*. = average daily residual for day t
N = number of observations
Finally, cumulative average residuals (CARs) will be calculated to
measure the total abnormal return accruing to the optionholders.
t
CAR^ = S U^
t=-4
The statistical significance will be measured by a t-test on
the daily residuals
40
^'t ' -^^
7 n (u. ^ - e)
1
. , N - 1
Residuals for the bidding and target firm will, of course, be treated
separately. The results are given in Table 3.3. For each day in the
event period, the daily average residual, t-statistic, probability (if
significant) of exceeding the absolute value of the t-statistic and
CAR are given.
Interpretation of results
It is interesting to note that with the possible exception of the
bidding firms' behavior on day-2, the results in Table 3.3 are consis-
tent with the results in Tables 3.1 and 3.2. Merger and acquisition
activity has a much larger impact on the target firm option holders
than the bidding firm optionholders. The cumulative average residual
is about 7.5% through the announcement day for the bidding firm options
versus about 39% for the target firm. Abnormal returns for the target
firm options are statistically significant two days and the day before
the announcement.
It would seem, however, that merger and acquisition activity is an
event for the bidding firm option holders. The excess return of 6.4%
two days before the announcement is highly significant. This is
consistent with the ISD behavior of the bidding firms' options on day-
2. Although not statistically significant, the ISD does change sign
and become positive (see Table 3.1). The issue of whether merger or
acquisition, should be regarded as an event (having measurable impact)
41
Table 3.3
Abnormal Returns in the Option Market
Around Merger and Acquisition Announcements
Bidding Firms
Daily
Day
Average
t-
Prob >
Car
Residual
Statistic
Itl
Dav-4
0.004993
0.21
0.004993
Day-3
0.007201
0.37
0.012194
Day-2
0.064110
2.87
0.0059
0.076304
Day-1
0.015931
0.40
0.092235
Day
-0.017628
-0.62
0.074607
Day+1
0.019649
0.74
0.094256
Day+2
0.022679
Tar
Daily
0.92
get Firms
0.116935
Day
Average
t-
Prob >
Car
Residual
Statistic
Iti
Day-4
-0.015935
-0.52
.
-0.015935
Day-3
0.033639
0.69
0.017704
Day-2
0.128971
1.75
0.0956
0.146675
Day-1
0.210493
2.68
0.0152
0.357168
Day
0.031266
0.45
0.388374
Day+1
0.017990
0.51
0.406364
Day+2
0.025705
-0.90
0.380659
42
on the bidding firm optionholders will be returned to in the next
section.
The above results, as might be expected, are consistent with the
hypothesis of market efficiency. For both the bidding and target
firms, the formal announcement is anticipated. After the merger and
acquisition is made public, there are no excess returns.
The "abnormal returns" in Table 4 are based upon the traditional
event study methodology that has been used in the equity market. That
is, the parameter(s) (beta in the equity market) for the model generat-
ing the predicted returns are estimated using data from some base
period free from the disturbing effects of the event (merger and
acquisition) activity. The difference between the actual, observed
market returns and the predicted returns is defined to be the excess or
abnormal return.
This excess return assumes that the risk (beta) does not change.
In actuality, merger and acquisition activity may not benefit a
stockholder even if abnormal returns are observed. These abnormal
returns may be accompanied by increased risk engendered by the merger
and acquisitionn activity. If risk were compensated for on a contin-
uous basis, it is possible that the abnormal returns reported would
disappear. This has not been done in the equity market since estimat-
ing beta requires time series data over a relatively lengthy period of
time. The issue is explored more fully in the next section.
For an event study in the option market, it is not necessary to
estimate beta. The relevant counterpart is the stock volatility for
which the ISD can be used as a proxy. The ISD, however, unlike beta,
can be computed at a point in time. This allows for a more complete
43
current estimate of predicted returns for event studies in the option
market than in the equity market.
In order to demonstrate this, the event study above was rerun for
the target firms on Day -2 and Day -1 (which yielded abnormal returns).
The only difference is that ISDs from the previous day (rather than 40
days prior to the announcement date) were used in the Black-Scholes
model to generate predicted returns. That is, prices for day-3 were
based on ISDs from day-4, prices for day-2 were based on prices from
day-3 and prices for day-1 were based on ISDs from day-2. All other
aspects of the study are identical. The results are shown in Table
3.4.
The implication of these results is that abnormal returns reported
in event studies to date may be overstated. Using the previous day's
ISDs to reflect a more current measure of the stock volatility reduced
the excess return on day-2 by almost three percentage points. Although
there is no direct relation between a stock's volatility and beta, it
would seem logical that merger and acquisition activity could have
short run effects. If beta could be observed on a continuous basis so
that equity returns could be properly adjusted for risk, abnormal
returns might be substantially reduced or even eliminated. This is
discussed in more detail in the next section.
The behavior of option markets around merger and acquisition
announcements
This section extends the event study in the option market
to the underlying equity. The reason for doing this is to compare the
behavior of the two markets around the announcement of merger and
acquisitions. There are two major reasons for doing this.
Table 3.4
Target Firm Option Abnormal Returns
Based On Previous Day ISDs
Daily
Day Average t- Prob >
Residual Statistic t
Day-2 0.099975 1.45 0.1617
Day-1 0.208600 2.49 0.0288
45
The first is that it places the option market results in perspec-
tive. While the absolute level of abnormal returns is of interest in
itself, it is important to compare the level of excess returns in the
option and equity markets. An investor concerned with merger and
acquisition activity would need to know the relative effects before he
could properly allocate his resources between the two markets.
The second reason for extending the event study to the underlying
stocks is that the two markets may behave differently. There are two
independent arguments for the hypothesis that merger and acquisition
activity will be first manifested in the option (rather than equity)
market.
Options can be interpreted as leveraged positions in the underly-
ing equity. The beta of an option is always greater than that of the
underlying asset (stock). Thus, it is possible that the option market
may be more sensitive to events than the equity market. In other
words, although both markets may have received the same bit of informa-
tion, the signal may be "magnified" and first apparent in the option
market .
It is also possible that the option market contains information
that is not incorporated in the equity market prior to major corporate
announcements. As mentioned previously, a call or put option can be
duplicated by an appropriate stock-bond portfolio. Because of this,
options have been viewed as "derivative" assets whose prices are
completely determined by the underlying equity. The possibility that
the option market may influence the equity market has received little
attention. Information may first be processed in the option market and
then filter to the equity market.
46
This issue has been investigated by Manaster and Rendleman
U982). They advanced the intriguing hypothesis that the option
market may play a key role in determining equilibrium stock prices.
They argue that some investors may prefer to invest in the option
rather than the equity market because of reduced transaction costs,
fewer short selling restrictions and most importantly, superior
leverage. These traders could push option prices out of equilibrium
relative to the underlying stocks. Arbitragers would then intervene
to restore equilibrium between the two markets.
Manaster and Rendleman attempted to test their theory. They
"inverted" the Black-Scholes model to solve for the implied stock
price. The implied stock price was then used to predict future stock
prices. They found some evidence that the option market contains
information that is not incorporated in the equity market. Unfortu-
nately, their results are very weak and fatally flawed by their
reliance on non-synchronous data. The data used in this dissertation
avoids this problem.
In retrospect, Manaster and Rendlemans' lack of results is
not surprising. Both the option and equity markets react to public
information. Generally, one would expect both markets to adjust
simultaneously to new public information. On any given day for any
particular corporation there may not be and probably is not information
that is not fully reflected in both markets.
However, this may not be true prior to m.ajor announcements by
corporations such as mergers or acquisitions. In this case, the option
market could be expected to be particularly influential in determining
stock prices. Keown and Pinkerton (1981) have argued that information
47
concerning impending mergers is susceptible to insider exploitation due
to the large number of people typically involved in the negotiating
process. They have attributed increased trading volume before the
merger announcement to insider activity. An insider attempting to
profit from knowledge of an impending merger would have an incentive to
use options because of their leverage aspects. To quote Fischer Black
(1975, p. 61), "Since an investor can usually get more action from a
given investment in options than he can be investing in the common
stock, he may choose to deal with options when he feels he has an
especially important piece of information." Option prices can be
expected ro contain more information than the equity market if non-
public information is being exploited. If information regarding future
mergers first reaches the financial markets through insider trading in
options, merger activity may very well be reflected in the option
market before the stock market.
Methodology and results
The standard event study methodology was applied to the equity
market. Daily returns for each day in the event period were obtained
from the Center for Security Price Research (CRSP) tapes. These
observed market returns were then compared to mean returns . Mean
returns were computed using returns for the sixty trading days prior to
the base date forty days prior to the announcement date.
Residual computation and analysis is as before. Residuals are
calculated for each company for each day in the event period
^'j't ~ ^i,t ' '^i
where U- ^ = residual for company j on day t
48
where U4 ^ = residual for company i on day t
R-j ^ = actual equity return for company j on day t
R-j = mean return for company j
Next daily average residuals are com.puted to measure the impact of the
merger or acquisition announcement for each day in the event period
_ n
U^ = 1/n E U.^.
3 = i
Cumulative average residuals are also calculated to measure the total
excess return accruing to the equityholder .
Statistical significance of the residuals is measured as before by
a t-test on the daily residuals.
U, • Vn
t =
n (U - e )2
v' E -^ r^-
n - i
j = l
The results are given in Table 3.5. For each day in the event period,
the daily average residual, t-statistic and probability of exceeding
the absolute value of the t-statistic, if significant, is given.
Table 3.5 is consistent with other merger studies done in the
equity market. Merger and acquisition activity has very little impact
on the bidding firms. The largest daily average residual, although
statistically significant at the 10% level is only 0.0044. For the
target firms, a statistically significant daily average return of
almost 0.04 was observed on day-1.
The abnormal returns for the target firm equityholders is a
little low compared to returns obtained in other merger studies. This
49
Table 3.5
Abnormal Returns In The Equity Market
Around Merger and Acquisition Announcements
Bidding Firms
Day
Day-4
Day-3
Day-2
Day-1
Day
Day+1
Day+2
Daily
Average
Residual
-0.001457
0.001275
0.004015
-0.002133
-0.007647
-0.003259
0.000562
t-
Prob >
atistic
/t/
Car
-0.65
-0.001457
0.50
-0,000182
1.71
0.0926
0.003833
-0.52
0.001700
-2.15
0.0362
-0.005947
1.26
-0.009206
0.22
-0.008694
Target
Daily
Average
t-
Day
Residual
Statistic
Dsy-4
-0.006512
-1.68
Day-3
-0.002720
-0.50
Day-2
0.007597
1.07
Day-1
0.039875
+2.44
Day
0.000620
0.08
Day+1
-0.002300
-0.36
Day+2
0.000402
0.08
Prob I
/t/
0.0240
Car
-0.006512
-0.008932
-0.001335
0.038540
0.039160
0.036860
0.037262
50
concerning impending mergers is susceptible to insider exploitation due
to the large number of people typically involved in the negotiating
process. They have attributed increased trading volume before the
merger announcement to insider activity. An insider attempting to
profit from knowledge of an impending merger would have an incentive to
use options because of their leverage aspects. To quote Fischer Black
(1975, p. 61), "Since an investor can usually get more action from a
given investment in options than he can be investing in the common
stock, he may choose to deal with options when he feels he has an
especially important piece of information." Option prices can be
expected to contain more information than the equity market if non-
public information is being exploited. If information regarding future
mergers first reaches the financial markets through insider trading in
options, merger activity may very well be reflected in the option
market before the stock market.
Methodology and results
The standard event study methodology was applied to the equity
market. Daily returns for each day in the event period were obtained
from the Center for Security Price Research (CRSP) tapes. These
observed market returns were then compared to mean returns. Mean
returns were computed using returns for the sixty trading days prior to
the base date forty days prior to the announcement date.
Residual computation and analysis is as before. Residuals are
calculated for each company for each day in the event period
"j't = Rj,t - Rj
51
is probably due to the sample. Companies listed on the CBOE tend to be
large, established companies. The takeover market may not be as
efficient for firms of this size. Relatively few companies have the
resources to undertake an acquisition of this scope. This fact is
reflected in the sample. Of the 21 target firms in the sample, seven
are mergers. For the divestitures, abnormal returns may also be
comparatively small due to the size of the firms involved. Although
large in absolute terms, a $100 million divestiture for a company such
as General Electric is likely to have very little impact.
Although the pattern of abnormal returns is similar for both
markets, the residuals in the option market tend to be much larger.
For the bidding firms, cumulative average residuals were 0.074607
through the announcement day for the options vice -0.005947 for the
equity. For the target firms, cumulative average residuals were
0.388374 and 0.039160 for the options and equity, respectively.
The only puzzling feature in the above tables is the statis-
tically significant excess return for the bidding firms options
observed two days prior to the announcement date. The results in the
equity market, however, are consistent. The average residual for day-
2, although small in absolute terms, is large compared to those of
other days and is statistically significant. It should be noted that
the data source used in the option and equity markets are independent.
The Berkeley option tapes served as the basis for the option event
study and the CRSP tapes for the equity.
The results for day-2 are also not due to low priced options. A
small price change on an option priced at less than a dollar could
result in large returns that m.ight not actually be realizable. This
52
possibility was checked for by redoing the analysis for the bidding
firm options on day-2 and the target firm options on day-1. This
time, however, rerurns based on prices less than one dollar are
eliminated. The results are given in Table 3.6. The daily average
residual for the bidding firm does decline from about 6.4% to roughly
A%. It is, however, still statistically significant. Eliminating the
low priced options from the target firms actually increases the daily
average residual.
The results from this study support the hypotheses that merger and
acquisition activity is first manifested in the option market. For the
bidding firms in the equity market, the daily average residual is
uniformly small. In the option market, however, there is a large jump
between the daily average residual of 0,007201 on day-3 and 0.06AI10 on
day-2. For the target firms, the evidence is more pronounced. In the
equity market, the merger and acquisition activity is not evident until
day-1. In the option market, the merger activity is definitely
reflected by the excess returns on day-2 and arguably on day-3. The
target firm ISDs, however, have started to react three days prior to
the announcement.
As noted earlier, abnormal returns were obtained for the target
firms in the equity market (see Table 3.5). These abnormal returns,
however, were based on historical data. Thus, an underlying assumption
is that the risk (beta) does not change. In reality, merger and
acquisition activity may be accompanied by increased risk that is not
reflected in the base beta. If beta could be observed on a continuous
basis so that equity returns could be properly adjusted for risk,
abnormal returns might be substantially reduced or even eliminated.
53
Table 3.6
Selected Abnormal Returns with Call Prices
Under $1.00 Eliminated
Bidding Fii
Day-2
■ms
Target Firms
Day-1
Daily
A/erage
Residual
t-
Statistic
Prob >
Itl
Daily
Average
Residual
t-
Statistic
Prob >
Itl
0.039593
1.98
0.0536
0.248540
2.76
0.0147
54
A number of attempts were made to adjust for risk in the equity
market by exploiting the high correlation between a stocks volatility
and beta. Unlike beta which requires time-series data, the ISD can be
calculated at a point in time. Although there is no theoretical
relationship between the ISD and a stock's beta, empirical relation-
ships can be established. These relationships can then be used to
adjust for the increasing risk due to the impending merger or acquisi-
tion announcement.
The methodology used to adjust the laevel of risk in the equity
market during the event period involves regressing stock betas against
their volatility. Daily returns for the target firms were regressed
against the market (CRSP value weighted) index for the six months
prior to the base (AO days prior to the announcement) date. This
yielded the intercept for the market model and an unadjusted beta to
conduct an event study in the equity market. Stock volatilities based
on the daily returns were also calculated. The target firm betas were
regressed against the volatilities to obtain the following relation
B = 0.866764 + 19.48914--'' a R^ = 0.098
A similar relationship was obtained using annual data. Annual
returns for the target firm were regressed against the market (CRSP
value weighted) index for the thirty years 1952 to 1981. Stock
volatilities based on this annual data were also computed. The annual
betas were then regressed against the stock volatilities to obtain
B = -0.061901 + 3.669232-- a R^ = 0.714
55
These relationships were used to adjust the beta for each day in
the event period. The ISD was plugged into the equations above to
obtain an adjusted beta. These adjusted betas were then plugged into
the market model (based on the daily returns) to generate predicted
returns. The standard event study methodology was then used to obtain
the results shown in Table 3.7. The first colmnn is the residuals and
associated t-statistics obtained from using an unadjusted beta, that
is, the beta based on the six months of daily data. The second column
shows the results obtained when the ISD is used to adjust the beta
using the relationship between beta and a based on the annual data.
The third coliamn shows the results when the ISD for each day in the
event period is used to adjust the beta using the relationship based on
the daily data.
These results indicate the adjustments for risk were not success-
ful. On day-1, the abnormal return are almost identical regardless of
whether the unadjusted beta, adjusted beta based on annual return data
or daily return data is used. Either the adjustment procedure is
flawed or the level of risk did not change during the event period. An
analysis of the data reveals a technical reason why the adjustment
procedure did not work.
In a CAPM framework, stocks must have an expected return greater
than the risk-free rate. Ex-post, however, negative returns do occur.
Many of the market returns on the day prior to the announcement date
(day-1) were negative in this sample. The average market return is
-0.000808. The practical effect of this is that adjusting beta upwards
can result in larger abnormal returns (residuals) because of the data.
If the market return is negative, increasing beta will only result in a
56
Table 3.7
Abnormal Returns for the Target Firms
for Various Beta Adiustments
Day Unadjusted Annually Daily
Beta Adjusted Adjusted
Beta Beta
-4 -0.002448 -0.003664 -0.003431
(t = -0.59) (t = -0.89) (t = -0.81)
-3 -0.002102 -0.001601 -0.001539
(t = -0.51) (t = -0.41) (t = -0.39)
-2 0.005530 0.005232 0.005516
(t = 0.84) (t = 0.79) (t = 0.82)
-1 0.032000 0.033300 0.033147
(t = 2.34) (t = 2.35) (t = 2.34)
Q 0.001017 0.001048 0.001032
(t = 0.13) (t = 0.14) (t = 0.14)
+1 -0.002665 -0.002597 -0.002729
(t = -0.66) (t = -0.64) (t = 0.26)
+2 0.001301 0.000929 0.001190
(t = 0.28) (t = 0.19) (t = 0.26)
57
lower predicted return. This illustrated by the abnormal returns that
result from the following adjustment to beta for day-1
Bg = B[(ISD(-1) - ISDb)/lSDb) + 1.0] • k
where B^ = adjusted beta
B = base beta obtained from six months daily data
ISD(-l) = ISD on day-1
ISDb = base ISD
k = an arbitrary scaler
The results for k = 1, 1,3, 1.5, and 2.0 are shown in Table 3.8. Here
we see that increasing beta has very little impact on the residuals. A
larger beta results in a larger predicted return (smaller residual)
impact for those companies for which the market return was positive.
This is offset, however, by those companies for which the market return
is negative.
The magnitude by which beta would have to be increased in order to
eliminated the abnormal returns can still be calculated. Adding 0.015
to the market returns on day-1 to make them positive and B-'.OIS to the
company returns does not change the residuals but makes the adjustment
process conform to theoretical expectations. The above regressions
were rerun with the indicated adjustment. The results are given in
Table 3.9. These results show that adjusting the base beta by the
percentage change in the ISDs times a scaler of 1.40 reduces the
abnormal returns to statistical insignificance. This suggests that the
basic methodology used to adjust beta above is sound but needs to be
applied to a larger sample where the average market return is positive.
58
Table 3.8
Abnormal Returns Obtained by Adjusting Beta by the
Percentage Change in ISDs Times a Scaler (K)
K = 1.0 K = 1.3 K = 1.5 K - 2.0
Day -1 0.033378 0.033793 0.034070 0.034751
(t - 2.34) (t = 2.35) (t = 2.35) (t = 2.33)
59
Table 3.9
Abnormal Returns Obtained by x'Vdjusting beta by the
Percentage Change in the ISD and a Scaler (K)
After Adding 0.015 to the Market Returns and
B"'.015 to the Company Returns
K = 1.0 K = 1.10 K = 1.20 K = 1.30 K = 1.40
Day-1 0.029318 0.027321 0.025323 0.023325 0.021327
(t = 2.19) (t = 2.05) (t = 1.91) (t = 1.77) (t = 1.62)
(0.0422) (0.0552) (0.0720) (0.0939) (12.17)
CHAPTER 4
VARIANCE BIAS AND NON- SYNCHRONOUS PRICES
IN THE BLACK- SCHOLES MODEL
One of the underlying assumptions of an event study in the equity
market is that the return generating process is linear. As long as
predicted returns equal observed (actual market) returns on average,
residuals (abnormal returns) should also average out to zero. It is
the common disturbance (event) that generates abnormal returns.
This linearity of the return generating process also justifies
using an estimate of beta. If an unbiasec estimator of beta is used,
errors will tend to offset. The estimates of beta may be high or low
but will average out in a large sample. Furthermore, the error in
estimated returns and thus residuals will also average out to zero.
The Black-Scholes model, however, is highly nonlinear. Thus,
using an estimate for the input variables may result in a systematic
bias. Even if an unbiased estimator is used for the input variables
(most notably the stock volatility), errors from the true call price
will not offset even in a large sample. The reason for this is that
equal deviations from the true input parameters will not result in
equal deviations from the true call price.
This has implications far beyond that of conducting an event study
in the options market. Applying the Black-Scholes model has an
inherent bias due to the fact that the formula is non- linear and input
variables must be estimated. The magnitude and direction of these
60
61
biases is of interest to any user of the Black-Scholes model. For this
reason, the issue of bias in the Black-Scholes model arising from these
sources is considered in a broader context rather than as a technical
issue concerning event study methodology.
This chapter has two sections. The first section deals with the
bias that results in the Black-Scholes model from using a sample
estimate of the variance with all other input parameters assumed to be
known. The following section extends the analysis to uncertainty in
the underlying stock price due to non-synchronous trading (or price
quotes) between the option and equity markets.
Variance Bias in the Black-Scholes Model
The Black-Scholes model is by far the most widely used option
pricing formula. In order to apply it, five input variables must be
obtained: the stock price, exercise price, time to maturity, risk-free
rate of interest and the volatility of the underlying stock. Of these
variables, four are directly observable. Only the variance of the
underlying stock returns needs to be estimated. Hull and White (1987)
have analyzed the impact of cp- itself being stochastic on the call
option value. In this paper, however, we assume that a^ is constant
but its estimate, S'^, is a random variable.
Classical methods of estimating the variance will bias the Black-
Scholes model as Ingersoll (1975) and Merton (1976) have pointed out.
To see this, define Z^ = Ln(l + R^) where R|. is the rate of return on
the underlying stock in period t and assijme that Z^ is an independent,
normally distributed random variable. The unbiased estimate, a^, of
the variance of the stock returns is given by
62
n
t=i
- Z)2
§2 =
=
(4.1)
N - 1
where N is the number of observations
_ n
Z = S Z^/N
t=l
While it is well known that S'^ is an unbiased estimate of a^, it is not
true that E(C) = C where C is the value derived from the Black-Scholes
formula with the true but unknown a^ and C is a random variable
calculated by employing the Black-Scholes formula with the random
variable S^.
Let us elaborate this point. The Black-Scholes model is given by
C = SgNCd^) - Ee'^tfj(^i2) ^ ^_2
where d^ = [InCSg/E) + (r + 0.5a2)t]/aVt
d2 = dj_ - aVt
and the sample estimate of C is given by C
C = SQN(d^) - Ee"^%(d2) ^-3
where dj^ = [InCSg/E) + (r 4- 0.5s2)t]/sV t
d2 = dj_ - sVt
(recall that S is a random variable)
It is obvious that E(C) ^ C for the following reasons. First, even
if E(s2) = a2, E(S) ^ a and a is one of the inputs into the Black-
Scholes formula. Second, even if E(S) = a (which it does not), E(C) ^
63
C since a appears in the denominator of the formula and E(l/S) ^ l/o.
Even if E(S) = a and E(l/S) = l/o, the model would still be biased due
to its non-linearity. Equal deviations from the the true a^ would not
result in equal deviations from the true option price.
Analyzing the gap between E(C) and C is difficult. One has to
evaluate the following difference
LnCSg/E) + (r+0.5s2)/sVt LnCSg/E) + (r-0. 5S)t/s2Vt
■O.SZ^ -rt -0.5Z2
E(C) - C = Sg J 1/V2TT e dz - Ee J 1/V2tt
e dz
LnCSo/E) + (r+0.5a2)t/aVt Ln(Sn/E) + (r-0.5o2)t/aVt
-0.5Z2 -rt -0.5Z2
-Sq J 1/V2TT e dz + Ee J 1/V2tt e dz
A closed-form solution to the first two integrals is extremely complex
since S, a random variable, appears in the upper bound. Boyle and
Anathanarayanan (1977) used numerical integration to approximate the
above integrals and investigated the case of an option expiring in 90
days.
In this paper, we provide an alternative approach by using
simulation. Sample estimates, S'-, of the stock volatility, o" ,
are generated and used to compute option prices using the Black-
Scholes formula. These prices are then compared to the theoretical
value determined by using the true o" in the Black-Scholes formula in
order to measure the bias induced. This is repeated for options with
64
various maturities. The dispersion of sample call prices from the
theoretical value is also investigated.
Methodology and Results
The effect of using a sample estimate of the variance in the
Black-Scholes model was analyzed using simulation analysis. For this
purpose, an option with the following characteristics was chosen.
These parameters were representative of IBM options in the early
1980 's. Note that the true stock volatility is assumed known.
Stock price = $68,125
Risk-Free Rate = 0.1325
of Interest
True Standard = 0.4472
Deviation of
Stock Returns
Time to Maturity = Various
Exercise Price = Various
The simulation is based on the well known relationship
'iTi'.veen the sample and true variance
9
S is distributed as a Chi-square with N-1 degrees of freedom which for
this analysis is assumed to be fifty-nine. This implies that the
sample variance was estimated using sixty observations. One thousand
Chi-square deviates were obtained using the International Mathematical
and Statistical Library (IMSL) computer program. The sample variance
was then computed for each Chi-square observation for input into the
Black-Scholes formula. For each exercise price, one thousand call
65
prices using the sample variances obtained from simulation were
calculated and the average computed. Options with five, sixty, and two
hundred seventy days to maturity were examined.
The results are given in Table 1. The theoretical price assxaming
the true variance is known is given for each exercise price. The
exercise price changes in five percent increments from the given stock
price of $68,125. The average simulation price is the mean of the
thousand generated call prices. The percentage bias is calculated by
% bias = '^'^h^o^^^tical price-average simulation price) * 100
theoretical price
Note that a positive bias is associated with average simulation prices
less than the theoretical Black-Scholes prices.
These results show a definite bias exists. While mean simulated
prices do deviate from theoretical Elack-Scholes prices, however, the
differences are small. In most cases, the average simulation price is
within a few cents of the theoretical price. The largest difference is
approximately eight cents. The percentage bias is also sm.all. For the
options with 60 and 270 days to expiration, it is always under one
percent. While biases over one percent do occur for the option with
five days to maturity, they are at prices so low as to be economically
meaningless .
A few observations on the nature of the bias between the average
simulation value and the theoretical value should be made. First, A
downward bias exists in most cases. The average value obtained from
simulation is less than the theoretical value for all nine exercise
prices for the options with sixty and two hundred seventy days to
expiration. For the five day option, the theoretical price exceeds the
66
Table 4.1
Theoretical Call Price and Average Simulation Call Price
T = 5 Days
to Maturity
Exercise
Theoretical
Average Simu-
Percent
Price
Price
lation Price
Bias
5A.500
13.7178
13.7157
0.01523
57.906
10.3184
10.3175
0.00872
61.313
6.9418
6.9427
-0.01253
64.719
3.7987
3.7966
0.05503
58.125
1.4799
1.4701
0.66084
71.531
0.3669
0.3636
0.89129
74.938
0.0549
0.0568
-3.49790
78.344
0.0050
0.0060
-20.16123
81.750
0.0003
0.0005
-50.71423
T = 60 Days to Maturity
Exercise
Theoretical
Average Simu-
Percent
Price
Price
lation Price
Bias
54.500
15.1807
15.1806
0.00040
57.906
12.3118
12.3038
0.06482
61.313
9.7241
9.7056
0.19035
64.719
7.4747
7.4468
0.37366
68.125
5.5917
5.5583
0.59802
71.531
4.0735
4.0402
0.81773
74.938
2.8926
2.8639
0.99112
78.344
2.0055
1.9846
1.04311
81.750
1.3596
1.3474
0.90314
T = 270 Days to Maturity
Exercise
Theoretical
Average Simu-
Percent
Price
Price
lation Price
Bias
54.500
20.9884
20.9614
0.12868
57.906
18.7896
18.7513
0.20132
61.313
16.7597
16.7107
0.29198
64.719
14.9000
14.8419
0.39013
68.125
13.2070
13.1416
0.49526
71.531
11.6748
11.6044
0.60293
74.938
10.2950
10.2220
0.70870
78.344
9.0591
8.9858
0.80924
81.750
7.9562
7.8849
0.89591
67
average simulation price for five exercise prices. In the other four
cases, the bias is extremely small amounting to less than one cent.
The bias is largest in absolute terms for the options with longer
maturities. However, there is no systematic relationship when the bias
is expressed in percentage terms. When the bias is expressed in
percentage terms, the bias for the sixty day option is smaller than
that of the two hundred seventy day option at low exercise prices but
larger at high relative exercise prices.
For an option of a given maturity, the bias is more pronounced at
high exercise prices. This holds true regardless of whether the bias
is expressed in absolute or percentage terms. This makes intuitive
sense. At low exercise prices most of an option's value is due to its
intrinsic worth. At high exercise prices more of the option's value
can be attributed to the volatility of the underlying stock. Conse-
quently, the estimate of the variance becomes more important.
The results in Table 4,1 are encouraging to users of the Black-
Scholes model. The bias in a large sample is small. This does not
guarantee, however, that using a sample estimate of the variance will
not severely degrade the applicability of the Black-Scholes model.
Sample call prices might each differ from the theoretical price by a
great amount. In a large sample these individual errors might offset
so that the average error was small. The dispersion of the sample call
prices from the theoretical value is also crucial.
For this reason, average simulated prices were generated on the
same IBM option as before only using A, 6, 8, 10, 15 and 30 runs
instead of a thousand. The results are given in Table 4.2. As before,
options with 5, 60 and 270 days to expiration were examined. For each
68
Table 4.2
Average Simulated Values and Bias for Various Sample Sizes
Maturity = 5 Days
Exercise Price = $54.50
Theoretical Price = $13.7178
Number
of
Average
Simulat
ions
Simulation Price
4
13.7178
6
13.7178
8
13.7178
10
13.7178
15
13.7178
30
13.7178
Percent
Bias
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Maturity = 5 Days
Exercise Price = $68,125
Theoretical Price = $1,4799
Number
of
Average
Simulat
ions
Simulation Price
4
1.5181
6
1.4626
8
1.4516
10
1.4526
15
1.4608
30
1.4669
Percent
Bias
-2.5812
1.1690
1.9123
1.8448
1.2906
0.8785
Maturity = 5 Days
Exercise Price = $81,750
Theoretical Price = $.0003
Number
of
Average
Percent
Simulat:
Lons
Simulation Price
Bias
4
0.0005
-66.6667
6
0.0004
-33.3333
8
0.0003
0.0000
10
0.0003
0.0000
15
0.0003
0.0000
30
0.0004
-33.3333
69
Table A. 2 (continued)
Maturity = 60 Days
Exercise Price = $54.50
Theoretical Price = $15.1807
Number
of
Average
Simulat
ions
Simulation Price
4
15.2367
6
15.1680
8
15.1528
10
15.1526
15
15.1618
30
15.1742
Percent
Bias
-0.3689
0.0837
0.1838
0.1851
0.1245
0.0428
Maturity = 60 Days
Exercise Price = $68,125
Theoretical Price = $5.5917
Numbe
r of
Average
Siinul
ations
Simulation Price
4
5. 7214
6
5.5328
8
5.4955
10
5.4980
15
5.5266
30
5.5473
Percent
Bias
-2.3195
1.0533
1.7204
1.6578
1.1642
0.7940
Maturity = 60 Days
Exercise Price = $81.75
Theoretical Price = $1.3596
Number of
Simulations
Average
Simulation
Price
Percent
Bias
4
6
8
10
15
30
1.4633
1.3239
1.2949
1.2962
1.3159
1.3355
-7.6272
2.6258
4.7588
4.6632
3.2142
1.7725
70
Table 4.2 (continued)
Maturity = 270 Days
Exercise Price = $54.50
Theoretical Price = $20.9884
Number
of
Average
Simulat
ions
Simulation Price
4
21.1675
6
20.9231
8
20.8727
10
20.8753
15
20.9101
30
20.9429
Percent
Bias
3.9112
0.3112
0.5513
0.5389
0.3731
0.2168
Maturity = 270 Days
Exercise Price = $68,125
Theoretical Price = $13.2070
Number
of
Average
Simulat
ions
Simulation Price
4
13.4635
6
13.0913
8
13.0177
10
13.0244
15
13.0789
30
13.1198
Percent
Bias
-1.9421
0.8760
1.4333
1.3826
0.9699
0.6603
Maturity = 270 Days
Exercise Price = $81.75
Theoretical Price = $7.9562
Number
of
Average
Simulat
ions
Simulation Price
4
8.2374
6
7.8293
8
7.7485
10
7.7559
15
7.8157
30
7.8606
Percent
Bias
-3.5344
1.5950
2.6105
2.5175
1.7671
1.2016
71
of these maturities, exercise prices of $54.50, $68,125 and $81.75 were
selected. The percent bias is calculated as before 5. The theoretical
Black-Scholes price is also given for each option.
These results show that the dispersion of option prices from their
theoretical values due to using the sample variance is not great. The
largest absolute difference is about $0.25. In general, the percentage
bias is usually less than 2%. The major exception is for the out-of-
the-money option with five days to maturity. This is due to the
insignificant theoretical call prices (less than $0.01) associated with
this option.
The same observations concerning the behavior of the bias for the
large sample (1000 runs) experiments apply to the small sample experi-
ments. The bias is generally positive (theoretical price exceeds
average simulated price). When the bias is negative, it is almost
always associated with the smallest number of simulations (4). Again,
the percentage bias is usually smallest at low exercise prices and
becomes larger as the exercise price is increased.
Non-synchronous Prices and the Black-Scholes Model
Many investment decisions involving options are based on closing
stock and option prices or other non-sychronous sources of data. Since
the option market is much thinner than the stock market, these prices
are often based on trades from different times of the day. The stock
price prevailing at the time of the last option trade may be signifi-
cantly different from the closing price at the end of the day. Conse-
quently, using this stock price in the Black-Scholes model may cause
options to appear mispriced as Trippi (1977), Chiras and Manaster
(1978), Galai (1977) and Bookstaber (1981) have pointed out.
72
Here the raispricing of options that can occur due to the non-
simultaneity or stock and option quotations and using a sample estimate
for the variance of the underlying stock returns in the Black-Scholes
model is examined. An option is constructed for analysis and its
theoretical value is calculated assuming the input variables including
the relevant stock price and true volatility a^, are known. This value
is compared to call prices generated with the same parameters (includ-
ing the true assumed volatility) only varying the input stock price
from the assumed true stock price in order to measure the effects of
nonsimultaneous stock and option quotations.
The additional bias resulting from using a sample estimate of the
variance is measured by simulation analysis. For each stock price,
sample estimates, 3--, of the stock volatility, o^ , are generated and
used to compute option prices using the Black-Scholes formula. These
prices are then compared to the theoretical price determined by using
the true variance, o^, and true (synchronous) stock price in order
measure the bias due to the combination of the two factors. The
methodology and results are describe below.
Methodology and Results
Simulation analysis was used to m.easure the effects in the
Black-Scholes model of using a sample estimate of the variance in
conjunction with nonsimultaneous stock and option quotations. For this
purpose, an option with the following characteristics was chosen. These
parameters are representative of a typical option traded on the Chicago
Board Option Exchange in the mid 1980 's. Note that the true stock
volatility and stock price are known.
73
Stock Price = $50,000
Risk-Free Rate = 0.1000
of Interest
True Standard = 0.3500
Deviation of
Stock Returns
Time to Maturity = Various
Exercise Price = $50,000
The effect of nonsiir.ultaneous stock and option prices alone on the
Black-Scholes model was measured by varying input stock price in 1/8
increments from the true stock price of $50,000. For each stock price
between $49,000 and $51,000 the Black-Scholes value was computed using
the parameters listed above including the true assumed variance of
0.3500.
The combined effects of nonsimultaneous price quotations and using
a sample estimate of the variance was analyzed by simulation. The
simulation is based on the relationship between the sample and true
variance didcusssed earlier
2
S N . 1 ^ A. 4
S^ is distributed as a Chi-square with N-1 degrees of freedom which for
this analysis is assumed to be twenty-nine. This implies that the
sample variance was estimated using thirty observations. One thousand
Chi-square deviates were obtained using the International Mathematical
and Statistical Library (IMSL) computer program. The sample variance
was then computed for each Chi-square observation for input into the
Black-Scholes formula. For each exercise price, one thousand call
prices using the sample variances obtained from simulation were
74
calculated and the average computed. Options with five, sixty, and two
hundred seventy days to maturity were examined.
The results are given in Table 4.3. For each stock price, the
Black-Scholes value is given based on the true variance of 0.3500.
This gives a measure of the mispricing that can occur to nonsimul-
taneous price quotations. For each of these stock prices, the average
simulation price is also given. The average simulated price is the
mean of the thousand generated call prices obtained with that exercise
price and estimates of the variance.
The percentage bias of these values from the theoretical value is
also given. The percentage bias is calculated by
% bias = ^^^^"^^^^^^^ price-average simulation price) * 100
theoretical price
Note that a positive bias is associated with average simulation prices
less than the theoretical Black-Scholes prices.
These results indicate that making investment decisions involving
options on the basis of nonsynchronous price data must be made with
great care. Even when the stock price is off by only an eighth the
observed call price will deviate from its theoretical price by over one
percent. For short maturities, using a stock price that deviates from
the true stock price by one dollar can results in call prices that are
over 50% off from the true value. The error due to using a sample
estimate of the true variance is small in comparison to that caused by
using nonconteraparenous stock prices. For stock prices above the true
value, the two errors are offsetting. For stock prices below the true
stock value, the two errors reinforce one another.
75
Table 4.3
Mispricing in the Black-Scholes Model Due to Nonsimultaneous
Stock and Option Quotations and Using a Sample Estimate
for the Variance of the Underlying Stock Returns
T = 5 Days to Maturity
Theoretical Price = 0.843231
Stock
Black-Scholes
Average
Percent Bias
Percent
Price
Price With
Simulation
Due to Non-
Bias Due to
True Variance
Price
Simultaneous
Quotations
Both Effects
51.000
1.455809
1.452143
-72.6465
-72.2117
50.875
1.369338
1.365278
-62.3918
-61.9103
50.750
1.285629
1.281119
-52.4646
-51.9298
50.625
1.2046Q5
1.199766
-42.8558
-42.2820
50.500
1.126509
1.121297
-33.5944
-32.9762
50.375
1.051239
1.045770
-24.6679
-24.0193
50.250
0.978912
0.973244
-16.0906
-15.4184
50.125
0.909561
0.903781
-7.8662
-7.1807
50.000
0.843231
0.837410
0.0000
-0.6903
49.875
0.779953
0.774134
7.5042
8.1943
49.750
0.719757
0.713988
14.6430
15.3271
49.625
0.662537
0.656968
21.4288
22.0892
49.500
0.608398
0.603052
27.8492
28.4832
49.375
0.557251
0.552217
33.9148
34.5117
49.250
0.509110
0.504414
39.6240
40.1809
49.125
0.463882
0.459599
44.9875
45.4954
49.000
0.417516
0.500117
50.0117
50.4644
76
Table 4.3 (continued)
T = 60 Days to Maturity
Theoretical Price = 3.135864
Stock
Black-Scholes
Average
Percent Bias
Percent
Price
Price With
Simulation
Due to Non-
Bias Due to
True Variance
Price
Simultaneous
Quotations
Both Effects
51.000
3.726805
3.708286
-18.8446
-18.2540
50.875
3.649992
3.631251
-16.3951
-15.7975
50.750
3.574020
3.555066
-13.9724
-13.3680
50.625
3.498869
3.479729
-11.5759
-10.9655
50.500
3.424468
3.405258
-9.2033
-8.5907
50.375
3.351058
3.331654
-6.8623
-6.2436
50.250
3.278503
3.258919
-4.5486
-3.9241
50.125
3.187067
3.187067
-2.2607
-1.6328
50.000
3.135864
3.116100
0.0000
0.6303
49.875
3.065887
3.046021
2.2315
2.8650
49.750
2.996780
2.976843
4.4353
5.0711
49.625
2.928482
2.908547
6.6132
7.2489
49.500
2.861113
2.841156
8.7617
9.3980
49.375
2.794615
2.774673
10.8821
11.5181
49.250
2.728973
2.709085
12.9754
13.6096
49.125
2.664169
2.644415
15.0419
15.6719
49.000
2.600462
2.580659
17.0735
17.7050
T = 270 Days to Maturity
Theoretical Value = 7.302824
Stock
Black-Scholes
Average
Percent Bias
Percent
Price
Price With
Simulation
Due to Non-
Bias Due to
True Variance
Price
Simultaneous
Quotations
Both Effects
51.000
7.948135
7.909295
-8.8364
-8.3046
50.875
7.866165
7.827100
-7.7140
-7.1791
50.750
7.784515
7.745256
-6.5959
-6.0584
50.625
7.703323
7.663871
-5.4842
-4.9439
50.500
7.622467
7.582793
-4.3677
-3.8337
50.375
7.541931
7.502134
-3.2742
-2.7292
50.250
7.461836
7.421933
-2.1774
-1.6310
50.125
7.382185
7.342033
-1.0867
-0.5369
50.000
7.302824
7.262574
0.0000
0.5512
49.875
7.223953
7.183511
1.0800
1.6338
49.750
7.145445
7.104862
2.1550
2.7108
49.625
7.067305
7.026569
3.2250
3.7829
49.500
5.989562
6.948733
4.2896
4.8487
49.375
6.912291
6.871284
5.3477
5.9092
49.250
6.835295
6.794231
6.4020
6.9643
49.125
6.758818
6.717631
7.4493
8.0132
49.000
6.682646
6.641387
8.4923
9.0573
77
Conclusion
In empirical tests of the black-Scholes model, one normally
employs ex-post estimates of o"^ since a^ itself is unknown. While the
sample variance is an unbiased estimate of a", the derived option value
(which is a random variable) is a biased estimate of the true Black-
Scholes value.
The effects of this bias were analyzed by simulation. The true
variance was assumed to be known and sample estimates generated by
using a Chi-square distribution. One thousand sample variances and
their associated call prices were obtained in each case. The average
call price was calculated and compared to the theoretical Black-Scholes
value. This process was performed on options with various maturities
and exercise prices.
The results show that using a sample estimate for the variance in
the Black-Scholes model results in a downward bias. The average
simulation price was less than the theoretical price for all options
with 60 and 270 days to maturity. For the 5 day option, the average
simulation price was less than the theoretical price for 5 of the 8
exercise prices. When an upward bias was observed, it was not economi-
cally significant. The downward bias was also evident in the small
sample experiments. Sample call prices were generated in the same
manner previously described only using fewer trials. Average simula-
tion prices were computed using 4, 6, 8, 10, 15 and 30 runs. The
average call prices generated by simulation were usually less than the
theoretical Black-Scholes price for six or more runs. The differences
between the average call prices generated by simulation and the
78
theoretical values were small. The percentage biases were also small
except for deep-out-of-the-money options close to expiration.
The effect of non- synchronous prices was also investigated. If
the input stock prices deviate from the true stock price by only 1/8,
the mispricing ranged from roughly 1% for the 270 day option to
approximately 7% for the 5 day option. The additional error due to
using an estimate of the variance was relatively small.
CHAPTER 5
SUMMARY AND CONCLUSIONS
This dissertation investigated the behavior of options around
merger and acquisition announcements. A variation of the traditional
event study methodology was applied to the option market in order to
determine the abnormal returns accruing to the bidding firm and target
firm optionholders. The event study was then extended to the under-
lying equity and the results between the two markets compared.
In both the equity and option market, the effect of merger and
acquisition activity was most pronounced for the target firms. The
cumulative average residuals for the bidding firms in the equity market
through the announcement date were close to zero. For the target
firms, they were close to 4%. The corresponding CARs in the option
market were 7.5% and 38.8%, respectively.
The abnormal returns for the target firms in the option market are
surprisingly large. Abnormal returns accruing to the optionholder are
over 10 times as large as those accruing to the equityholders . The is
due is not only the leverage effect in options but the fact that the
stock volatility is increasing as well.
Merger and acquisition activity can be expected to have a larger
impact on the volatility (second moment of the return generating
function) of the target firms than of the equity firms. Event studies
in the equity market have shown that most of the gains from merger
activity are captured by the target firm shareholders. The rationale
79
80
for this is that the takeover market is competitive. If a company has
some unique aspect to exploit, it will find or have the potential to
find a number of bidders. Competition among the bidding firms will
drive the net present value of the investment to zero.
Because of this, merger and acquisition activity should be
expected to affect the volatility of the target firms' much more than
the bidding firms'. Merger and acquisition activity is a more or less
neutral event for the bidding firm shareholders. Target firm
shareholders are much more likely to be greatly affected. The
importance of merger and acquisition activity combined with uncertainty
over the terms of the agreement, whether alternative bidders will
appear, whether the agreement will be consummated, etc., should result
in higher ISDs for the target firms.
This hypothesis was confirmed. The change in the ISDs between the
event period and base date for the bidding firms was not significant.
The changes were small and statistically insignificant. The target
firms, however, had large statistically significant changes in the
ISDs.
The effect of changing stock volatility on option prices was also
examined. Option prices in the event period were compared to those
using the Black-Scholes model using the current stock price but the
base ISD. The results showed that changing stock volatility was an
important factor in the abnormal returns reaped by the target firm
optionholders.
The results of this study also suggest that merger and acquisition
activity is first reflected in the option market. The target firm
ISDs started to increase and were statistically significant 3 days
81
before the announcement. Target firm option returns had started to
increase and were statistically significant 2 days before the announce-
ment. Target firm stock returns, however, did not significantly
increase until the day prior to the announcement. Bidding firm option
returns were economically and statistically significant two days before
the announcement. Bidding firm stock returns were very small through
the announcement date although they were statistically significant 2
days before the announcement.
These results have practical implications for investors. If
someone anticipates a company is about to announce a merger or acquisi-
tion, they would reap much greater returns by purchasing options rather
than the stock. Furthermore, they would be substantially better off by
purchasing the target firm option rather than that of the bidding firm.
This dissertation analyzed two issues involving the event study
methodology. The first was the proper adjustment for risk in the
equity market. Predicted returns have usually been based on data from
some base period. The traditional event study methodology, thus,
implicitly assumes that risk remains constant. It is far more likely
that risk is actually changing due to the event (merger and acquisition
activity). Abnormal returns would thus be overstated. Empirical
relationships between the ISD (stock volatility) and beta were
developed. These relationships were then used to adjust beta during
the event period. Although conceptually sound, the results were
disappointing due to a technical factor. The market return for many of
the companies was negative. This resulted in smaller predicted returns
(larger residuals) when larger betas were plugged into the market
model.
82
The second major issue involves event studies in the option
market. The Black-Scholes model is non-linear. Unbiased estimators
for the input variables will still bias the results since equal
deviations from the true input parameter value will not result in equal
deviations from the true call price. Simulation analysis was used to
measure the magnitude of this effect. The results indicate that
although caution must be used in interpreting the results of an event
study that uses the Black-Scholes model to generate predicted returns,
the error is usually small.
ij REFERENCES
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87
BIOGRAPHICAL SKETCH
James A. Yoder was born on June 18, 1953, at Fort Monmouth, New
Jersey. He received his Bachelor of Science degree in mathematics in
197A from the State University of New York at Albany. He then went on
to obtain an M.A. in economics in 1975 from the same university.
Mr. Yoder entered the Navy in the Nuclear Power Program. He
served on board the U.S.S. Dwight D. Eisenhower and made one
Mediterranean deployment. After leaving the Navy, he completed his MBA
at the State University of New York at Albany before entering the Ph.D.
program at the University of Florida.
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
1 <
u
c/-"^..'^'^"
1
Haim Levy, Chairman-''
Walter J. Mather ly Professor of
Finance
T ce-rtify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy. /
/t,...
Roy Cium
Professor of Finance, Insurance,
and Real Estate
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Sanfor/'V. Berg ^^
Profei^'sor of Economics
This dissertation was submitted to the Graduate Faculty of the Depart-
ment of Finance, Insurance, and Real Estate in the College of Business
Administration and to the Graduate School and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy-
Dean, Graduate School
December 1988